In the bowl of a stand mixer fitted with the dough hook, add flour, sugar, yeast, and spice.
Turn mixer on low to mix dry ingredients.
Add milk, butter, egg, egg yolk, and vanilla.
Mix on low speed until ingredients are combined.
Increase speed to medium.
Knead until dough is smooth and pulling away from the bowl, about 8-10 minutes.
Transfer dough to a container that has been sprayed with cooking spray.
Cover.
Let rise in a warm place, or place in a proofing box set to 78℉ with the water tray filled, until doubled in bulk, about 1 hour.
Turn dough out onto a solid surface. (You shouldn't need flour; but if you do, dust lightly.)
Roll dough to ⅜ inch thick.
Use a 3 inch cutter to cut out doughnuts. (You can re-roll the scraps.)
Place on a piece of parchment sprayed lightly with cooking spray.
Cover with plastic wrap.
Let proof for 15 minutes.
Pour 2 inches of vegetable oil in a dutch oven or large saucepan.
At the end of the 15 minutes, heat your oil to 355℉-360℉/179℃-182℃. (Total proof time for the doughnuts is 30-45 minutes.)
Fry doughnuts 1 or 2 at a time, until puffed and golden, about 1 minute.
Flip and cook on other side until puffed and golden, about 1 minute.
Drain.